Approximating Random Variables by Stochastic Integrals

Let X be a semimartingale and Θ the space of all predictable X-integrable processes ϑ such that ∫ϑ dX is in the space S² of semimartingales. We consider the problem of approximating a given random variable H ² by a stochastic integral ∫T₀ ϑₛ dXₛ, with respect to the L²-norm. If X is special and has the form X = X₀ + M + ∫ α d M, we construct a solution in feedback form under the assumptions that ∫ α² d M is deterministic and that H admits a strong F-S decomposition into a constant, a stochastic integral of X and a martingale part orthogonal to M. We provide sufficient conditions for the existence of such a decomposition, and we give several applications to quadratic optimization problems arising in financial mathematics.